The no-hiding theorem[1] states that if information is lost from a system via decoherence, then it moves to the subspace of the environment and it cannot remain in the correlation between the system and the environment. This is a fundamental consequence of the linearity and unitarity of quantum mechanics. Thus, information is never lost. This has implications in black hole information paradox and in fact any process that tends to lose information completely. The no-hiding theorem is robust to imperfection in the physical process that seemingly destroys the original information.
This was proved by Samuel L. Braunstein and Arun K. Pati in 2007. In 2011, the no-hiding theorem was experimentally tested[2] using nuclear magnetic resonance devices where a single qubit undergoes complete randomization; i.e., a pure state transforms to a random mixed state. Subsequently, the lost information has been recovered from the ancilla qubits using suitable local unitary transformation only in the environment Hilbert space in accordance with the no-hiding theorem. This experiment for the first time demonstrated the conservation of quantum information.[3]
In physics, conservation laws play important roles. For example, the law of conservation of energy states that the energy of a closed system must remain constant. It can neither increase nor decrease without coming in contact with an external system. If we consider the whole universe as a closed system, the total amount of energy always remains the same. However, the form of energy keeps changing. One may wonder if there is any such law for the conservation of information. In the classical world, information can be copied and deleted perfectly. In the quantum world, however, the conservation of quantum information should mean that information cannot be created nor destroyed. This concept stems from two fundamental theorems of quantum mechanics: the no-cloning theorem and the no-deleting theorem. But the no-hiding theorem is a more general proof of conservation of quantum information which originates from the proof of conservation of wave function in quantum theory. It may be noted that the conservation of entropy holds for a quantum system undergoing unitary time evolution and if entropy represents information in quantum theory, then it is believed then that information should somehow be conserved. For example, one can prove that pure states remain pure states and probabilistic combination of pure states (called as mixed states) remain mixed states under unitary evolution. However, it was never proved that if the probability amplitude disappears from one system, it will reappear in another system. Now, using the no-hiding theorem one can make a precise statement. One may say that as energy keeps changing its form, the wave function keep moving from one Hilbert space to another Hilbert space. Since the wave function contains all the relevant information about a physical system, the conservation of wave function is tantamount to conservation of quantum information.